Continuity of volumes on arithmetic varieties

TL;DR

This paper introduces a new volume function for hermitian invertible sheaves on arithmetic varieties and proves its continuity, leading to several important applications in arithmetic geometry.

Contribution

It defines the arithmetic volume function and establishes its continuity, providing new tools and results in the study of arithmetic varieties.

Findings

Proves the continuity of the arithmetic volume function.

Derives the arithmetic Hilbert-Samuel formula for nef sheaves.

Establishes applications like the generalized Hodge index theorem and Bogomolov-Gieseker inequality.

Abstract

We introduce the volume function for hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic Hilbert-Samuel formula for a nef hermitian invertible sheaf. We also give another several applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker's inequality, etc.